On Bohr's theorem for general Dirichlet series

We present quantitative versions of Bohr's theorem on general Dirichlet series D=∑ane−λns assuming different assumptions on the frequency λ=(λn) , including the conditions introduced by Bohr and Landau. Therefore, using the summation method by typical (first) means invented by M. Riesz, without any condition on λ, we give upper bounds for the norm of the partial sum operator SN(D):=∑n=1Nan(D)e−λns of length N on the space D∞ext(λ) of all somewhere convergent λ‐Dirichlet series, which allow a holomorphic and bounded extension to the open right half plane [Re>0] . As a consequence for some classes of λ's we obtain a Montel theorem in D∞(λ) ; the space of all D∈D∞ext(λ) which converge on [Re>0] . Moreover, following the ideas of Neder we give a construction of frequencies λ for which D∞(λ) fails to be complete.


INTRODUCTION
A general Dirichlet series is a formal sum ∑ − , where ( ) are complex coefficients (called Dirichlet coefficients), a complex variable and ∶= ( ) a strictly increasing nonnegative real sequence tending to +∞ (called frequency). To see first examples we choose = (log ) and obtain ordinary Dirichlet series ∑ − , whereas the choice = ( ) = (0, 1, 2, …) leads to formal power series ∑ regarding the substitution = − . Within the last two decades the theory of ordinary Dirichlet series had a sort of renaissance which in particular led to the solution of some long-standing problems (see [11] and [20] for more information). A fundamental object in these investigations is given by the Banach space  ∞ of all ordinary Dirichlet series ∶= ∑ − , which converge and define bounded functions on [ > 0]. One of the main tools in this theory is the fact that every ordinary Dirichlet series ∈  ∞ converges uniformly on [ > ] for all > 0, which is a consequence of what is called Bohr's theorem and was proven by Bohr in [4].
Several years ago in [1] this "ordinary" result was improved by a quantitative version. An important consequence is that Bohr's theorem implies that  ∞ is a Banach space (see [11, §1.4]). The natural domain of Bohr's theorem for general Dirichlet series is the space  ∞ ( ) of all somewhere convergent -Dirichlet series = ∑ − allowing a holomorphic and bounded extension to [ > 0]. Additionally we define the subspace  ∞ ( ) of all ∈  ∞ ( ) which converge on [ > 0]. Notice that with this notation we have  ∞ ((log )) =  ∞ . The inclusion  ∞ ( ) ⊂  ∞ ( ) in general is strict (see e.g. the frequencies constructed in [19, §1]). A natural norm on  ∞ ( ) (and on  ∞ ( )) is given by ‖ ‖ ∞ ∶= sup [ >0] | ( )|, where is the (unique) extension of . Note that a priori, ‖ ⋅ ‖ ∞ is only a semi norm, that is it could be possible for a particular Dirichlet series = ∑ − with some ≠ 0 to have a bounded holomorphic extension with ‖ ‖ ∞ = 0, or equivalently, it is not clear whether  ∞ ( ) can be considered as a subspace of ∞ [ > 0], the Banach space of all holomorphic and bounded functions on [ > 0]. Here it is important to distinguish Dirichlet series from their limit function, and to prove that ‖ ⋅ ‖ ∞ in fact is a norm on  ∞ ( ) requires to check that all Dirichlet coefficients of vanish provided ‖ ‖ ∞ = 0 (see Corollary 3.9).
We say that a frequency satisfies Bohr's theorem (or Bohr's theorem holds for ) if every ∈  ∞ ( ) converges uniformly on [ > ] for all > 0. It was a prominent question in the beginning of the 20th century for which 's Bohr's theorem holds.
Actually Bohr proves his theorem not only for the case = (log ) but for the class of 's satisfying the following condition (we call it Bohr's condition ( )): roughly speaking this condition prevents the 's from getting too close too fast. Then in [4] Bohr shows that if satisfies ( ), then Bohr's theorem holds for . Note that = (log ) satisfies ( ) with = 1.
In [18] Landau gives another sufficient condition (we call it Landau's condition ( )), which is weaker than ( ) and extends the class of frequencies for which Bohr's theorem holds: We like to mention that in [19, §1] Neder considered 's satisfying and then proved that this condition is not sufficient for satisfying Bohr's theorem by constructing, giving some > 0, a Dirichlet series (belonging to some frequency ) for which ( ) = ( ) = and ( ) ≤ 0 hold. In particular this shows that the inclusion  ∞ ( ) ⊂  ∞ ( ) is strict for these 's.
Like Bohr, Landau under his condition ( ) only proves the qualitative version of Bohr's theorem. Of course, establishing quantitative versions means to control the norm of the partial sum operator Then using the summation method of typical means of order > 0 invented by M. Riesz (Proposition 3.4), our main result gives an estimate of ‖ ‖ without assuming any condition on (Theorem 3.2).
Main result. For all 0 < ≤ 1 and ∈ we have where Γ is the Gamma function and > 0 a universal constant.
As a consequence assuming Bohr's condition (1.2) on the choice ∶= 1 , ≥ 2 (since 1 = 0 is possible), leads to Another application of the summation method of typical means gives an alternative proof of the fact that -linearly independent 's (that is ∑ = 0 implies = 0 for all finite rational sequences = ( )) satisfy Bohr's theorem, which was proven by Bohr in [6]. More precisely we show that in this case the space  ∞ ( ) equals 1 (as Banach spaces) via ∑ − ↦ ( ) (Theorem 4.7). Moreover, we would like to consider  ∞ ( ) as a Banach space. Unfortunately it may fail to be complete. Based on ideas of Neder we give a construction of 's for which  ∞ ( ) is not complete (Theorem 5.2). But there are sufficient conditions on , including ( ) and -linearly independence, we present in Theorem 5.1.
Before we start let us mention that recently in [8] given a frequency the authors introduced the space  ∞ ( ) of all series of the form ∑ − , which converge and define a bounded function on [ > 0]. Then defining ∶= ( log ( )) we have  ∞ ( ) =  ∞ ( ) and so both approaches are equivalent in this sense. All results on  ∞ ( ) in [8] are based on the assumption that satisfies the condition ( ). In contrast to this article, we here try to avoid assumptions on as much as possible.
This text is inspired by the work of (in alphabetical order) Besicovitch, Bohr, Hardy, Landau, Neder, Perron, and M. Riesz. In Section 3 we prove our main result and in Section 4 we apply it to obtain quantitative variants of Bohr's theorem under different assumptions on , including ( ) and ( ). We finish by Section 5, where we face completeness of  ∞ ( ). We start recalling some basics on Dirichlet series.

GENERAL DIRICHLET SERIES
As already mentioned in the introduction a strictly increasing nonnegative real sequence ∶= ( ) tending to +∞ we call a frequency. Then general Dirichlet series = ∑ − belonging to some we call -Dirichlet series and we define ( ) to be the space of all (formal) -Dirichlet series. Moreover, the (complex) coefficient is called the th Dirichlet coefficients of . Additionally we define, provided ( ) < ∞, ( ) = inf{ ∈ | the limit function of allows a holomorphic and bounded extension to [ > ]}.

A Bohr-Cahen formula
where again the equality holds if the left hand side is nonnegative. In this section we derive (2.1) from the following proposition concerning uniform convergence of sequences of Dirichlet series and take advantage of both in Section 4. Then the particular case of a sequence of partial sums will reprove (2.1).
Therefore given a sequence of (formal) -Dirichlet series = ∑ − we define where we endow 2 with the product order, that is ( , ) ≤ ( , ) if and only if ≤ and ≤ .

MAIN RESULT AND APPROXIMATION BY TYPICAL RIESZ MEANS
Recall that by Definition 2.3 a frequency satisfies Bohr's theorem if every ∈  ∞ ( ) converges uniformly on [ > ] for all > 0 or equivalently the equality holds for all somewhere convergent -Dirichlet series. As already mentioned in the introduction it was a prominent question in the beginning of the 20th century for which 's the equality (3.1) holds. The following remark shows how control of the norm of the partial sum operator As announced, our main result gives bounds of ‖ ‖ without any assumptions on , which is a sort of uniform version of [13, Thm. 21, p. 36].

Theorem 3.2. For all
where > 0 is a universal constant and Γ denotes the Gamma function.
Proposition 3.4 is indicated after the proof of [13, Thm. 41, p. 53] (without inequality (3.2)). In the language of [13] it states that on every smaller halfplane [ > ] the limit functions of Dirichlet series ∈  ∞ ( ) are uniform limits of their typical (first) means of any order > 0. The proof relies on a formula of Perron (see [13, Thm. 39, p. 50]). We give an alternative proof of this formula (Lemma 3.6) using the Fourier inversion formula and we first deduce (3.2) from it. Then, using a Bohr-Cahen type formula for the abscissa of uniform summability by typical first means (Lemma 3.8), we show that (3.2) implies the first part of Proposition 3.4. Note that satisfies Bohr's theorem, if the first part of Proposition 3.4 is valid for = 0.
The second main ingredient for the proof of Theorem 3.2 links partial sums of a Dirichlet series to its typical means.
For the sake of completeness we like to mention, that actually, if = 0 and = for some , then (3.3) also holds true, if the integral on the right hand side is defined by its principle value (see [13,Thm. 13,p. 12]). This case (which is not needed in the following) is not covered by our alternative proof of Lemma 3.6, where we use the Fourier inversion formula. We need the following observation. . Then for all ∈ [ > ] (see [13,Thm. 24,p. 39], where > 0 is considered, but the case = 0 follows in the same way applying Abel summation) and so where ∉ if = 0 and is arbitrary if > 0. This implies ( where equality holds whenever ( ) is nonnegative.
Proof. Let denote the right hand side of (3.7). We first show (3.7) and assume that < ∞, since otherwise the claim is trivial. Hence, fixing > 0, there is a constant such that for all ‖ ‖ ‖ ‖ ( ) ‖ ‖ ‖ ‖∞ ≤ ( + ) . In particular, with the choice = and = , where ∈ , by (3.8) the first term on the right hand side of (3.9) vanishes whenever → ∞. Applying integration by parts we obtain where the second derivative appearing is given by and so for all > 0 with 2 = 2 ( ) = ( which tends to zero uniformly in as → ∞. Analogously, using which also vanishes uniformly in tending → ∞. It remains to consider the integral with 1 . The dominated convergence theorem implies for all and we claim that the convergence is uniform in . Indeed, we have the calculation we used before (in particular using the adapted variant of (3.11)) we obtain By substitution with = we for every , , > 0 obtain In particular, if 0 < < 1, then the choice = and = 1 − gives (3.14) Using that, we continue estimating Hence for all > 0 which finally completes the proof.
Proof. Since on [ = ] the limit function is the uniform limit of 1 ( ) tending → ∞ (Proposition 3.4), is almost periodic. Then by [3, Ch. I, §3.11, p. 21] we have Another property of almost periodic functions is that they allow a unique continuous extension to the Bohr compactification of (see [20, §1.5.2.2, Thm. 1.5.5]). In particular, the monomials − ⋅ extend uniquely to characters on . We like to mention that this observation led to an  -theory of general Dirichlet series (see [9]) naturally containing and extending the  -theory of ordinary Dirichlet series invented by Bayart in [2].
To complete our proof of Theorem 3.2, it remains to verify Lemma 3.5.
Proof of Lemma 3.5. We again use (3.6) and obtain for all ∈ Now, applying [13, Lemma 7, p. 28] to the real and imaginary parts of the integral we obtain and so

ON BOHR'S THEOREM
Now we apply our main result Theorem 3.2 to prove quantitative variants of Bohr's theorem for certain classes of 's (including ( ) and ( )) by giving bounds for ‖ ‖, ∈ . Observe that by Corollary 3.9 we always have the trivial bound ‖ ‖ ≤ . Hence by Remark 3.1 satisfies Bohr's theorem (or equivalently equality For instance = ( ) = (0, 1, 2, …) fulfils (( )) = 0 and we (again) see as a consequence that for power series we cannot distinguish between uniform convergence and boundedness of the limit function up to . We like to mention that the number ( ) also has a geometric meaning. Bohr shows in [5, §3, Hilfssatz 3, Hilfssatz 2] that where the latter is the maximal width of the so called strip of pointwise and not absolutely convergence.

Bohr's condition
We already know from Theorem 4.2 that if ( ) holds for , then satisfies Bohr's theorem, since ( ) implies ( ) (Remark 4.1). But the stronger assumption ( ) improves the bound for the norm of .
To put it differently the conditions ( ) and ( ) states that the sequence (log( 1 +1 − )) increases at most linearly respectively exponentially, and the quality of the growth gives different bounds for ‖ ‖. We consider now 's whose growth is somewhat in between: Proof of Theorem 4.4. As before w.l.o.g. we assume that +1 − ≤ 1. Then choosing = 1 , ≥ 2, we obtain (with

-linearly independent frequencies
In [6] Bohr proves that -linearly independent 's satisfy the equality = for all somewhere convergent -Dirichlet series. In this section we give an alternative proof to Bohr's using Proposition 3.4 and the so-called Kronecker's theorem, which states that the set {( − ) | ∈ } is dense in ∞ , whenever the real sequence ( ) is -linearly independent. The latter is equivalent to the fact that for every choice of complex coefficient  where ∶= ( ) (the th Taylor coefficient of ) whenever = in its prime number decomposition. This identification links the space of all -homogeneous Dirichlet series  ( ) ∞ ((log )) to the space of -homogenous polynomials (or equivalently bounded -linear forms) on 0 (see [11, §2, §3]). In particular,  , where is is the th prime number, is -linearly independent, Theorem 4.7 recovers this result.

A Montel theorem
In [2, Lemma 18] Bayart proves that every bounded sequence ( ) ⊂  ∞ (log ) allows a subsequence ( ) and some ∈  ∞ ((log )) such that ( ) converges uniformly to on [ > ] for all > 0; a fact which is called "Montel theorem" and extends to the following classes of 's.

ABOUT COMPLETENESS
Recall that from Corollary 3.9 we know that (  ∞ ( ), ‖ ⋅ ‖ ∞ ) is a normed space. In this section we face completeness. We first state sufficient conditions on for completeness of  ∞ ( ) and  ∞ ( ). Then we give a construction of 's for which  ∞ ( ) fails to be complete. We like to mention that in [8] it is already proven that ( ) is sufficient for completeness of  ∞ ( ) by introducing the following condition, which is equivalent to ( ): (1) is -linearly independent, (2) ( ) = 0,